Mastering Completing the Square | 配方法完全攻略

📚 Mastering Completing the Square | 配方法完全攻略

Completing the square is one of the most powerful algebraic techniques in IGCSE Mathematics. It allows you to rewrite a quadratic expression in a form that reveals its turning point, axis of symmetry, and maximum or minimum value. This skill is essential for solving quadratic equations, sketching graphs, and solving problems involving optimisation.

配方法是 IGCSE 数学中最强大的代数技巧之一。通过配方法，我们可以将二次表达式改写为能够直接显示顶点、对称轴以及最大值或最小值的形式。这一技能在解二次方程、绘制函数图像以及解决最优化问题时都不可或缺。

1. What Does “Completing the Square” Mean? | 什么是配方法？

Completing the square is the process of rewriting a quadratic expression of the form ax² + bx + c into the form a(x − h)² + k. The key idea is to create a perfect square trinomial from the x² and x terms, then adjust with a constant term. The resulting form clearly shows the turning point (h, k) of the parabola.

配方法是将形如 ax² + bx + c 的二次表达式改写为 a(x − h)² + k 的过程。核心思想是从 x² 项和 x 项中构造出一个完全平方三项式，再用常数项进行调整。改写后的形式能直接显示抛物线的顶点坐标 (h, k)。

For example, the expression x² + 6x + 5 can be rewritten as (x + 3)² − 4. This tells us immediately that the vertex of the graph y = x² + 6x + 5 is at (−3, −4).

例如，表达式 x² + 6x + 5 可以改写为 (x + 3)² − 4。这让我们立刻知道函数 y = x² + 6x + 5 的图像顶点在 (−3, −4)。

2. The Basic Formula | 基本公式

For a quadratic in the form x² + bx + c (where the coefficient of x² is 1), completing the square gives:

对于形如 x² + bx + c 的二次式（即 x² 的系数为 1），配方结果为：

x² + bx + c = (x + b/2)² − (b/2)² + c

Notice that we take half of b, use it inside the bracket, and then subtract its square outside to keep the expression equivalent. This is the single most important formula to memorise.

注意，我们取 b 的一半放入括号内，然后在括号外减去它的平方，以保证表达式与原式等价。这是最需要牢记的一个公式。

For example, x² + 8x + 3. Here b = 8, so half of b is 4. We write:

例如，对于 x² + 8x + 3，此时 b = 8，b 的一半是 4。我们写为：

x² + 8x + 3 = (x + 4)² − 16 + 3 = (x + 4)² − 13

3. When the Coefficient of x² Is Not 1 | 当 x² 的系数不为 1 时

If the quadratic is written as ax² + bx + c with a ≠ 1, we first factor out a from the first two terms, then complete the square inside the bracket. Let us use 2x² + 8x + 5 as an example.

如果二次式为 ax² + bx + c 且 a ≠ 1，我们首先从前两项中提出 a，再在括号内配方。以 2x² + 8x + 5 为例。

Step 1: Factor out 2 from the first two terms:

第一步：从前两项中提出 2：

2x² + 8x + 5 = 2(x² + 4x) + 5

Step 2: Complete the square for x² + 4x:

第二步：对 x² + 4x 配方：

2[(x + 2)² − 4] + 5

Step 3: Distribute and simplify:

第三步：展开并化简：

2(x + 2)² − 8 + 5 = 2(x + 2)² − 3

4. Solving Quadratic Equations by Completing the Square | 用配方法解二次方程

Completing the square can also solve quadratic equations. We first rewrite the quadratic expression in completed-square form, then isolate the squared term and take the square root of both sides. Remember to include the ± sign.

配方法也可以用来解二次方程。我们先将二次表达式改写为配方形式，然后分离出平方项，对方程两边同时开平方。切记要加上 ± 号。

Example: Solve x² + 6x − 7 = 0 by completing the square.

例题：用配方法解方程 x² + 6x − 7 = 0。

Step 1: Complete the square on the left-hand side:

第一步：对方程左边配方：

(x + 3)² − 9 − 7 = 0 → (x + 3)² − 16 = 0

Step 2: Move the constant to the other side:

第二步：将常数移到等号另一边：

(x + 3)² = 16

Step 3: Take square roots:

第三步：两边开平方：

x + 3 = ±4

Step 4: Solve for x:

第四步：解出 x：

x = 1 or x = −7

5. Finding the Turning Point of a Parabola | 求抛物线的顶点

When a quadratic function is written in the completed-square form y = a(x − h)² + k, the turning point of the parabola is exactly at (h, k). If a > 0, the turning point is a minimum; if a < 0, it is a maximum.

当二次函数写成配方法形式 y = a(x − h)² + k 时，抛物线的顶点就在 (h, k)。如果 a > 0，顶点为最小值点；如果 a < 0，顶点为最大值点。

Example: Find the turning point of y = 2x² + 8x + 5.

例题：求 y = 2x² + 8x + 5 的顶点。

From Section 3, we already found:

从第 3 节我们已经得到：

y = 2(x + 2)² − 3

Comparing with y = a(x − h)² + k, we have h = −2 and k = −3. Therefore the turning point is (−2, −3), and since a = 2 > 0, this is a minimum point.

与 y = a(x − h)² + k 对比，得到 h = −2，k = −3。因此顶点为 (−2, −3)，由于 a = 2 > 0，这是最小值点。

6. The Line of Symmetry | 对称轴

The line of symmetry of a parabola is always the vertical line passing through the turning point. In completed-square form y = a(x − h)² + k, the line of symmetry is simply x = h.

抛物线的对称轴总是经过顶点的竖直线。在配方形式 y = a(x − h)² + k 中，对称轴就是 x = h。

For the example above, y = 2(x + 2)² − 3, the line of symmetry is x = −2. This means the graph is a mirror image of itself on the two sides of this vertical line.

对于上述例子 y = 2(x + 2)² − 3，对称轴为 x = −2。这意味着图像在该竖直线两侧互为镜像。

If you have the graph of a quadratic, you can also find the axis of symmetry by averaging the two x-coordinates of the x-intercepts. But when you have the completed-square form, the process is immediate.

如果你有二次函数的图像，也可以通过两个 x 轴交点的 x 坐标取平均值来找到对称轴。但当你已有配方形式时，可以直接读出对称轴。

7. Minimum and Maximum Values | 最大值与最小值

The completed-square form makes it trivial to find the minimum or maximum value of a quadratic function. Since a square is always non-negative, the k value is the minimum (if a > 0) or the maximum (if a < 0) of the function.

配方形式让我们可以轻松找到二次函数的最小值或最大值。因为平方项始终非负，所以当 a > 0 时 k 为最小值，当 a < 0 时 k 为最大值。

For y = 2(x + 2)² − 3, the minimum value is −3. This occurs when (x + 2)² = 0, i.e., when x = −2.

对于 y = 2(x + 2)² − 3，最小值为 −3。这发生在 (x + 2)² = 0 时，即 x = −2。

For y = −(x − 1)² + 5, the maximum value is 5, occurring at x = 1.

对于 y = −(x − 1)² + 5，最大值为 5，发生在 x = 1 处。

This idea connects directly to real-world problems such as finding the maximum profit or minimum cost where a quadratic model is used.

这个想法直接联系到实际应用问题，例如在二次模型中寻找最大利润或最小成本。

8. Sketching Quadratic Graphs | 绘制二次函数图像

Completing the square gives you the key features needed to sketch a quadratic graph accurately. You now know the turning point, the line of symmetry, and whether the parabola opens upward or downward. To complete the sketch, you may also find the y-intercept and x-intercepts.

配方法为你提供了准确绘制二次函数图像所需的关键特征。现在你知道顶点、对称轴以及抛物线的开口方向。为了完整绘图，你还可以求出 y 截距和 x 截距。

Let us summarise the steps using the function y = x² − 4x + 3.

我们用函数 y = x² − 4x + 3 来总结绘图步骤。

First, complete the square:

首先，配方：

y = (x − 2)² − 4 + 3 = (x − 2)² − 1

Key features:

关键特征：

Turning point: (2, −1), minimum because a = 1 > 0
顶点：(2, −1)，最小值，因为 a = 1 > 0
Line of symmetry: x = 2
对称轴：x = 2
y-intercept: set x = 0 → y = 3
y 截距：令 x = 0 → y = 3
x-intercepts: set y = 0 → (x − 2)² = 1 → x = 1 or x = 3
x 截距：令 y = 0 → (x − 2)² = 1 → x = 1 或 x = 3

Now you have enough information to produce an accurate sketch of the parabola.

现在你已拥有足够的信息来准确绘制抛物线草图。

9. Useful Formula for a Quadratic in General Form | 一般式的快捷公式

For the general quadratic y = ax² + bx + c, you can memorise a shortcut for finding the completed-square form directly. But it is often safer to apply the step-by-step method each time to avoid algebraic errors.

对于一般式 y = ax² + bx + c，可以记忆一个直接得到配方形式的快捷公式。但为了减少代数错误，更稳妥的方法是每次都按步骤来。

The x-coordinate of the turning point is given by x = −b/(2a). To find the y-coordinate, substitute this x back into the original expression.

顶点的 x 坐标为 x = −b/(2a)。要求 y 坐标，只需将这个 x 代回原表达式。

Let us verify with our earlier example y = 2x² + 8x + 5. Here a = 2, b = 8, so x = −8/(2 × 2) = −2. Substituting x = −2 gives:

用之前的例子 y = 2x² + 8x + 5 来验证。这里 a = 2，b = 8，所以 x = −8/(2 × 2) = −2。将 x = −2 代入原式得：

y = 2(−2)² + 8(−2) + 5 = 8 − 16 + 5 = −3

This matches our earlier result, where the turning point was (−2, −3). Using this formula can save time in a multiple-choice setting, while the step-by-step method is more transparent for showing your work.

这与之前的结果一致：顶点为 (−2, −3)。在选择题中，这个公式可以节省时间；而在需要展示过程时，逐步法更为清晰。

10. Common Pitfalls and How to Avoid Them | 常见错误与避免方法

Many students make avoidable mistakes when completing the square. Let us identify the most frequent ones and explain how to prevent them.

许多学生在配方时会犯一些可以避免的错误。我们来识别最常见的几个并说明如何预防。

Forgetting the sign inside the bracket: The expression x² − 6x becomes (x − 3)², not (x + 3)². Always take half of b with its sign.
忘记括号内的符号：x² − 6x 应变为 (x − 3)²，而不是 (x + 3)²。要始终带上 b 的符号取一半。
Not subtracting (b/2)² correctly: When completing the square for x² + bx + c, you must subtract (b/2)², not add it.
没有正确减去 (b/2)²：对 x² + bx + c 配方时，必须减去 (b/2)²，而不是加上它。
Forgetting to factor out a when a ≠ 1: If the coefficient of x² is not 1, you must factor it out before completing the square, then multiply back carefully.
当 a ≠ 1 时忘记提出 a：如果 x² 的系数不为 1，必须先提出该系数，配方后再小心乘回去。
Missing the ± sign when solving: When taking the square root of both sides of an equation, always write ±.
解方程时漏写 ± 号：对方程两边开平方时，永远要写 ±。

Check your result by expanding the completed-square form back to the original quadratic. This verification step takes only a few seconds and catches most errors.

可以通过将配方形式展开，检验是否变回原二次式来验证结果。这个检验步骤只需几秒钟，却能发现大多数错误。

11. Worked Exam-Style Question | 典型考试题精讲

Let us now work through a full exam-style question that requires several skills at once.

现在我们完整解一道需要综合运用多种技能的考试风格题目。

Question: The function f(x) = 2x² − 12x + 7 is defined for all real x.

题目：已知函数 f(x) = 2x² − 12x + 7 对所有实数 x 均有定义。

(a) Express f(x) in the form a(x − h)² + k.

(a) 将 f(x) 化为 a(x − h)² + k 的形式。

Solution: Factor out 2 from the first two terms:

解答：从前两项中提出 2：

2(x² − 6x) + 7

Complete the square inside:

在括号内配方：

2[(x − 3)² − 9] + 7 = 2(x − 3)² − 18 + 7 = 2(x − 3)² − 11

(b) State the minimum value of f(x) and the value of x at which it occurs.

(b) 指出 f(x) 的最小值以及取得最小值时的 x 值。

Solution: Since a = 2 > 0, the function has a minimum. The minimum value is k = −11, occurring at x = h = 3.

解答：因为 a = 2 > 0，函数有最小值。最小值为 k = −11，在 x = h = 3 时取得。

Solution: Set the completed-square form to zero:

解答：令配方形式等于零：

2(x − 3)² − 11 = 0 → (x − 3)² = 11/2

x − 3 = ±√(11/2) → x = 3 ± √(11/2)

This can also be written as x = 3 ± (√22)/2 by rationalising the denominator, but either form is acceptable in IGCSE exams.

通过有理化分母，也可以写成 x = 3 ± (√22)/2，两种形式在 IGCSE 考试中均可接受。

12. Practice Questions | 练习巩固

Use the following exercises to test your understanding. Complete the square for each expression or equation, and check your answers by expanding back.

用以下练习来检验你的理解。对每个表达式或方程进行配方，并通过展开回来检查答案。

Question \| 题目	Answer \| 答案
1. x² + 10x + 23	(x + 5)² − 2
2. x² − 4x − 1	(x − 2)² − 5
3. 3x² + 6x + 1	3(x + 1)² − 2
4. −x² + 8x − 7	−(x − 4)² + 9
5. Solve x² + 5x + 2 = 0	x = (−5 ± √17)/2

If you struggled with any of these, review the relevant section above before moving on. Completing the square requires practice, but once mastered, it will make many other topics much easier.

如果这些题目让你感到困难，请先回头复习相关章节再继续。配方法需要反复练习，但一旦掌握，会让许多其他主题变得更加容易。

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Mastering Completing the Square | 配方法完全攻略